1. Field of the Invention
This invention relates to the analysis of characteristics of an overall structure consisting of a plurality of sub structures coupled together and, more particularly, to a system for and a method of analyzing characteristics of an overall structure such as a structure vibration simulation system, which analyzes vibrations of a plurality of structures, either theoretically by a finite element method utilizing a computer or experimentally by using a conventional FFT analyzer used for vibration analysis, and estimates vibration characteristics of an overall structure obtained by coupling together these structures by using a computer before actually manufacturing such overall structure.
2. Description of the Prior Art
An overall structure characteristics analyzer will now be described in conjunction with a structure vibration simulation system. Computer-aided engineering (CAE), which is used in the design of structures, particularly machine structures for performing modeling and simulation of structures with a computer before trial manufacture, has been attracting attention as a powerful means for reducing development time and cost. In the CAE, vibration analysis is important as a reliability evaluation method in the design of machine structures. As prior art methods of vibration analysis of machine structures, there are an experimental FFT (fast Fourier transform) analysis method and a theoretical finite element analysis method. Further, there is a sub structure synthesis method (or a building block approach) as disclosed in Japanese Patent Application 63-060766, in which the experimental FFT analysis and theoretical analysis based on the finite element method are performed with respect to each element (sub structure) of a machine structure to be analyzed, and the results of the analyses are used to numerically simulate the vibration characteristics of the machine structure (overall structure).
In the experimental FFT analysis method, the operations of causing vibrations of a machine structure, measuring responses at this time, sampling these signals with an A-D converter, supplying sampled digital data to a minicomputer or microcomputer, performing FFT of the data and producing a transfer function between the vibration application point and the response point, are performed repeatedly for various points of the machine structure, and modal parameters such as peculiar vibration frequency of the structure, damping ratio and vibration mode by means of curve fitting (or modal analysis). This method is used as an important means for obtaining the vibration characteristics of the actual structure.
The finite element method, on the other hand, is a method of theoretical analysis utilizing a computer. In this method, a machine structure is thought to be capable of being expressed by a collection of a finite number of finite elements. A relation between an externally applied force and a resultant deformation is obtained for each element. These relations are used to define a displacement function concerning the relation between an externally applied force and a resultant deformation of the overall machine structure. Using this displacement function, a stiffness matrix [K] and a mass matrix [M] are obtained to solve an eigen value problem given as EQU [M]{x}+[K]{x}={o} (1)
where {x} represents a displacement vector, and . . . represents a second order time differential. Also, the peculiar vibration frequency and vibration mode of the structure are obtained. Further, an equation of motion given as EQU [M]{x}+[C]{x}+[K]{x}={f} (2)
where [M] represents a mass matrix. [C] an damping matrix, [C]=.alpha.[M]+.beta.[K], .alpha. and .beta. damping ratios, [K] a stiffness matrix, {f} external force, {x}, {x} and {x} displacement, velocity and acceleration vectors, is solved to obtain the response analysis of each element.
The sub structure synthesis method is one in which experimental FFT analysis and theoretical analysis based on the finite element method are performed with respect to each element (or sub structure) of the machine structure for analysis, and results of the analyses are numerically simulated. A specific example of this method will now be described with reference to FIG. 9.
FIG. 9 illustrates a simulation concerning a railway car design. Referring to the Figure, reference numeral 100 designates a car body, 101 a chassis, and 102 and 103 local bases A and B. These parts constitute elements of the railway car. Designated at 110 to 113 are examples of vibration characteristic of car body 100, chassis 101 and local bases A 102 and B 103, respectively, and 121 is an example of vibration characteristic of overall system obtained by sub structure synthesis method 120. In the graphs of the vibration characteristic examples, the ordinate x is taken for the vibration response, and the abscissa f is taken for the frequency. Designated at 200 is a co-ordinate system, in which the railway car is found.
This co-ordinate system represents a three-dimensional co-ordinate space defined by perpendicular x, y and z axes. Designated at 11 to 14, 21 to 24 and 31 to 34 are points of measurement selected in car body 100 and chassis 101. Designated at A and B are selected points of measurement in chassis 101 and local bases A 102 and B 103. Points of measurement designated by like reference numerals or symbols constitute a point of coupling when the individual elements are coupled together. Generally, vibration response at one point of measurement may be examined by considering the following six different directions as shown in co-ordinate system 200:
(1) Direction (x) of the x axis, PA1 (2) Direction (y) of the y axis, PA1 (3) Direction (z) of the z axis, PA1 (4) Direction (p) of rotation about the x axis, PA1 (5) Direction (q) of rotation about the y axis, and PA1 (6) Direction (r) of rotation about the z axis. PA1 [H.sub.rr ]: represents a transfer function matrix H.sup.(1) requiring no co-ordinate conversion, PA1 [.GAMMA.]: represents a constraint relation matrix for co-ordinate conversion, PA1 {X.sub.d }: represents a displacement vector of a degree of freedom after co-ordinate conversion, PA1 {X.sub.i }: represents a displacement vector of a degree of freedom before co-ordinate conversion, PA1 {X.sub.r }: represents another displacement vector of a degree of freedom, PA1 {F.sub.d }: represents an external force vector of a degree of freedom as the subject of co-ordinate conversion, and PA1 {F.sub.r }: represents another external force vector of a degree of freedom.
These directions are referred to as degrees of freedom. Thus, there are at most six degrees of freedom at one point of measurement. In a system in which the directions p, q and r of rotation can be ignored, there are only three degrees x, y and z of freedom. Further, where only a spring undergoes a vertical motion, there is only a single degree of freedom (in the sole direction x, for instance).
Now, the determination of a transfer function which is extensively used for analyzing vibration characteristics of elements will be described.
If 12 points 11 to 34 of measurement in car body 100 each have three degrees x to z of freedom, there is a total of N=12.times.3=36 degrees of freedom.
In this way, one or more points of measurement with a total of N degrees of freedom are selected in a structure, with numerals 1, 2, . . . N provided to designate the individual degrees of freedom, and by setting a degree m of freedom to be a direction of response and another degree l of freedom to be a direction of vibration application, a vibration of a predetermined waveform (the vibration being expressed as displacement, velocity or acceleration of the pertinent point of measurement) is applied in the direction of vibration application, and vibration in the direction of response is measured.
In this case, the frequency spectrum of vibrations in the direction of vibration application is well known, and as for the frequency spectrum of vibration in the direction of vibration application, the vibration transfer function H between the direction l of pressure application and the direction m of response can be expressed as a function H m, l(.omega.) of angular frequency .omega. to determine H m, .sub.1 (.omega.), H m, .sub.2 (.omega.), . . . , H m, .sub.N (.omega.). Further, H m, m (.omega.)=1 and H m, l (.omega.)=H l, m(.omega.), which is referred to as the theorum of reciprocity.
This transfer function H m , (.omega.) is set as an N.times.N matrix
to obtain a transfer function matrix ##EQU1## with respect to the sub structure, given as ##STR1##
Then, by denoting a force fl applied in each direction l of vibration application, an external force vector F.sup.(k) with respect to sub structure k in directions 1 to N of vibration application is expressed as ##EQU2##
An equation of motion expressed by a transfer function for each local structure is EQU [G.sup.(k) ]{F.sup.(k) }={x.sup.(k) } (5) ##EQU3##
The transfer function calculated here concerns the transfer function matrix ##EQU4## of compliance (displacement/force), and for conversion to the transfer function matrix ##EQU5## of dynamic stiffness (force/displacement), it is converted to the transfer function of dynamic stiffness by obtaining the inverse matrix by using an equation ##EQU6##
In this way, an equation of motion given as ##EQU7## is given for each local structure.
Now, a method of obtaining an overall system equation by coupling together the individual sub structures will be described.
First, a method of obtaining a system by coupling together two sub structures and producing an equation of motion of the system will be described. Of the two sub structures, the degrees of freedom are classified to be those ({xm.sup.(1) }, {xm.sup.(2) }) where a further sub is coupled and those ({xr.sup.(1) }, {xr.sup.(2) }) where no further sub structure is coupled. For example, xm.sup.(1) and xm.sup.(2) may be thought to be the degrees of freedom of points of measurement which show the same response when and only when sub structures are coupled together by bolting. The equations of motion of individual the sub structures are given as ##EQU8##
When {xm.sup.(1) } and {xm.sup.(2) } are coupled together, the equation of motion of the individual sub structures are now ##EQU9## where {P} represents force applied by sub structure 1 to sub structure 2 and {-p} represents force applied by sub structure 2 to sub structure 1. By removing p by using equations (7) and (11) and a coupling condition EQU {xm.sup.(1) }={xm.sup.(2) }(=xm}) (12)
we obtain ##EQU10## This equation is an equation of motion of the system obtained by coupling together the two sub structures. Here, {Fm} is {Fm}={Fm.sup.(1) }+{Fm.sup.(2) } and represents an external force acting on coupling point {xm}. The method of producing the equation of motion of the system obtained by coupling together two sub substructures can be seen by expressing the coefficient matrix of equation (13) as ##STR2##